Applications of the Householder transformation to ridge-type estimation methods
type: Item
instance of: scholarly article
title: Applications of the Householder transformation to ridge-type estimation methods
zbMATH Open document ID: 0760.65135
DOI: 10.1016/0096-3003(92)90071-8
author: David A. Cicci
published in: Applied Mathematics and Computation
publication date: 1993-01-17T00:00:00Z
review text: The author considers the model \(y=Hx+e\), where \(y\) is an \((m\times 1)\) vector of observation residuals, \(H\) is an \((m\times n)\) mapping matrix, \(x\) is an \((n\times 1)\) state vector of corrections to a reference solution, \(e\) is an \((m\times 1)\) vector of observation errors, \(m\) is the total number of observations and \(n\) is the number of model variables in the state vector. The observations are assumed to have a Gaussian distribution with zero mean and covariance equal to \(R\). Instead of estimating the unknown \(x\) by the classical least squares estimator \(\hat x=(H^ TR^{-1}H)^{-1}H^ TR^{-1}y\) the author proposes to use \(\hat x^*=(H^ TR^{-1}H+D_ R^{-1}KD^{-1}_ R)^{-1}H^ TR^{-1}y\), where \(D\) is an \((n\times n)\) diagonal matrix whose \(i\)th diagonal element is the reciprocal of the square root of the corresponding element of \(H^ TR^{-1}H\) and \(K\) is an \((n\times n)\) diagonal matrix of biasing parameters. The reason is to stabilize the computation in the case when \(H^ TR^{-1}H\) is ill-conditioned. The main result of the paper is a detailed algorithm describing Householder transformations which can be used in this modification in order to obtain desired \(QR\)-decomposition which can be useful in the modified situation.
Mathematics Subject Classification ID: 65C99
Mathematics Subject Classification ID: 62J07
Mathematics Subject Classification ID: 65F25
zbMATH DE Number: 96924
zbMATH Keywords: least squares estimator
zbMATH Keywords: ridge regression
zbMATH Keywords: linear model
zbMATH Keywords: \(QR\)- decomposition
zbMATH Keywords: Householder transformations
zbMATH Keywords: ill-conditioned
reviewed by: Jaromír Antoch
MaRDI profile type: Publication
full work available at URL: 0096-3003(92)90071-8
OpenAlex ID: W2044586181
cites work: Ridge Regression: Biased Estimation for Nonorthogonal Problems
cites work: Ridge Regression: Applications to Nonorthogonal Problems
cites work: Q5180217
cites work: Unitary Triangularization of a Nonsymmetric Matrix
cites work: Q5588184
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description: scientific article; zbMATH DE number 96924